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Math Help - Gradient and Divergent Identities

  1. #1
    Senior Member bugatti79's Avatar
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    Gradient and Divergent Identities

    Folks,

    I need to show that \int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds given

    \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds where \Omega and \Gamma are the domain and boundary respectively. F,G and w are scalar functions...any ideas?

    I attempted to expand the LHS but I didnt feel it was leading me anywhere...

    \displaystyle \int_\Omega (\hat{e_x}\frac{\partial G}{\partial x}+\hat{e_y}\frac{\partial G}{\partial y})w dx dy....?
    Last edited by bugatti79; July 12th 2012 at 10:56 AM. Reason: typo
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  2. #2
    Senior Member bugatti79's Avatar
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    Re: Gradient and Divergent Identities

    If we let u=w## then ##du=dw=\nabla w???

    \displaystyle dv=\nabla G dxdy then

    \displaystyle v=\int_\Omega \nabla G dxdy=\int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})Gds

    Thus

    \displaystyle \int_\Omega(\nabla G)wdxdy= \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G w ds- \int \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G \nabla w ds

    Clearly I have gone wrong somewhere....? Thanks


    Note: This has been posted at PF approx week and a half ago. Unlikely to be answered at this stage. Gradient and Divergent Identities
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